Unified characterization for higher-order topological phase transitions

Higher-order topological phase transitions (HOTPTs) are associated with closing either the bulk energy gap (type-I) or boundary energy gap (type-II) without changing symmetry. The type-I and type-II transitions are usually characterized separately and are hard to distinguish in experiments. Here we propose a unified topological characterization of the HOTPTs, which further enables a precise detection of the both types of topological transitions by quench dynamics. For a higher-order topological phase a correspondence is shown between the mass domain walls on real-space boundaries and the higher-order band-inversion surfaces (BISs) which are certain interfaces in the momentum subspace. The topological phase transitions occur when momentum-space topological nodes, dubbed higher-order topological charges, cross the higher-order BISs after proper projection. Particularly, the bulk (or boundary) gap closes when all (or part of) topological charges cross the BISs, characterizing the type-I (or type-II) HOTPTs. We finally show that our unified characterization can be easily measured from quench dynamics, which is driven with control in experiments.